CF: Continued fraction convergents

Description

Returns continued fraction convergent using the modified Lenz's algorithm; function CF() deals with continued fractions and GCF() deals with generalized continued fractions.

Usage

CF(a, finite = FALSE, tol=0)
GCF(a,b, b0=0, finite = FALSE, tol=0)

Arguments

a,b

In function CF(), the elements of a are the partial denominators; in GCF() the elements of a are the partial numerators and the elements of b the partial denominators

finite

Boolean, with default FALSE meaning to iterate Lenz's algorithm until convergence (a warning is given if the sequence has not converged); and TRUE meaning to evaluate the finite continued fraction

In function GCF(), floor of the continued fraction

tol

tolerance, with default 0 silently replaced with .Machine$double.eps

Details

Function CF() treats the first element of its argument as the integer part of the convergent.

Function CF() is a wrapper for GCF(); it includes special dispensation for infinite values (in which case the value of the appropriate finite CF is returned).

The implementation is in C; the real and complex cases are treated separately in the interests of efficiency.

The algorithm terminates when the convergence criterion is achieved irrespective of the value of finite.

References

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling 1992. Numerical recipes 3rd edition: the art of scientific computing. Cambridge University Press; section 5.2 “Evaluation of continued fractions”
W. J. Lenz 1976. Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics, 15(3):668-671

Examples

Run this code

# NOT RUN {
phi <- (sqrt(5)+1)/2
phi_cf <- CF(rep(1,100))     # phi = [1;1,1,1,1,1,...]
phi - phi_cf     # should be small

# The tan function:
"tan_cf" <- function(z,n=20){
     GCF(c(z, rep(-z^2,n-1)), seq(from=1,by=2, len=n))
}

z <- 1+1i
tan(z) - tan_cf(z)   # should be small

# approximate real numbers with continued fraction:
as_cf(pi)

as_cf(exp(1),25)    # OK up to element 21 (which should be 14)

  # Some convergents of pi:
  jj <- convergents(c(3,7,15,1,292))
  jj$A / jj$B - pi


  # An identity of Euler's:
  jj <- GCF(a=seq(from=2,by=2,len=30), b=seq(from=3,by=2,len=30), b0=1) 
  jj - 1/(exp(0.5)-1)   # should be small

# }